The first error certainly sounds like a version mismatch. The APR library is a runtime library based on your path (in itself, it can be based on where you are running Tomcat). It looks like you have an APR library for Tomcat 6. Perhaps Eclipse provides this APR library.
Tomcat comes with an APR library, but to facilitate installation, Tomcat does not automatically change the path to a point in the APR library. Integration Eclipse Tomcat can (I'm not sure about this, haven't used integration before), including the APR libraries, to make Tomcat easier to use.
Check and make sure that Eclipse is not configured to use an earlier version of Tomcat, and if so, either upgrade Eclipse to Tomcat 7, or downgrade Tomcat to the version specified in Eclipse.
If you cannot understand this, then you do not have to worry about it at the moment. Tomcat should work anyway without the APR libraries. The only benefit you get from the APR libraries is increased productivity, as this is probably the beginning of your project, which does not need to be developed immediately.
The second problem, / manager / page, might be the Tomcat configuration. Tomcat does not actually display the manager page (and this will give you this error), unless users are configured with administrator privileges .
Try just / (i.e. http: // localhost: 8080 ), this page should load regardless of user configuration.